Problem 67 - Algebra 2

Question

Write an exponential equation \(y=a b^{x}\) whose graph passes through the given points. \(\left(0, \frac{1}{3}\right)\) and \((2,3)\)

Step-by-Step Solution

Answer

The exponential equation of the graph that passes through points \((0, \frac{1}{3})\) and \((2,3)\) is \(y = 3^x\).

Step 1: Plug in the coordinates of the first point

Plug the coordinates of the first point \((0, \frac{1}{3})\) into \(y=a b^{x}\) to get \(\frac{1}{3}=a b^{0}\). Any number (except 0) raised to the power of 0 is 1, which gives you \(a = \frac{1}{3}\).

Step 2: Plug in the coordinates of the second point

Now, substitute \(a = \frac{1}{3}\) and the second coordinates\((2,3)\) into our equation to solve for 'b'. Thus, \(3=(\frac{1}{3}) b^2\). By multiplying both sides by 3 and taking square root we get \(b=3\).

Step 3: Formulate the exponential equation

Now that we have the values for both 'a' and 'b', we can substitute them back into the given equation. So the equation becomes \(y=\frac{1}{3}3^x\), or simply \(y = 3^x\).

Key Concepts

Coordinate Points

In the context of graphs and equations, coordinate points are used to identify specific locations on a 2-dimensional plane. Each point is represented by a pair of numbers, written in parentheses as \(x, y\). The first number is the "x-coordinate," which tells you how far across the plane the point is, while the second number is the "y-coordinate," indicating how far up or down the point is.

• The x-coordinate represents the horizontal distance from the origin (0,0).
• The y-coordinate represents the vertical distance from the origin.
• When a graph passes through given points, it means when you substitute the x-value in the equation, the y-value that results is the coordinate's y-value.

For the exercise given, our coordinate points were \(0, \frac{1}{3}\) and \(2, 3\). This tells us two things happen on the graph: when \(x = 0\), \(y = \frac{1}{3}\); and when \(x = 2\), \(y = 3\). Understanding how to read these points is crucial for determining the equation that fits the graph.

Solving Equations

Solving equations means finding the value(s) of the variable(s) that make the equation true. Let's break down the strategy used in the original solution.

• Firstly, substitute the known coordinates into the equation. This helps determine one part of the unknown equation. For instance, plugging the point \(0, \frac{1}{3}\) into the equation \(y = a b^{x}\) reveals that \(a = \frac{1}{3}\).
• Then, use the second coordinate to solve for the remaining variable. Substitute the value of \(a\) into the same equation with the second coordinate to find \(b\). For example, using \(2, 3\), we solved \(3 = (\frac{1}{3}) b^2\) for \(b = 3\).
• Reconstruct the equation using the values found. In this case, substituting back both \(a\) and \(b\) gave us the equation \(y = 3^x\).

This method allows us to securely form the exponential equation that matches the graph passing through the given points.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. This can be written in the form \(y = a b^{x}\), where 'a' is a constant that scales the function vertically, 'b' is the base that determines the growth rate, and 'x' is the exponent or input variable.

• In our specific problem, we discovered that \(a = \frac{1}{3}\) and \(b = 3\), forming the function \(y = \frac{1}{3} 3^{x}\), which simplifies to \(y = 3^x\).
• This shows that for every unit increase in 'x', the value of \(y\) is multiplied by 3, highlighting rapid exponential growth.
• Exponential functions are essential in various fields such as science and finance, because they effectively model processes involving growth or decay over time.

Understanding exponential functions is critical in interpreting and plotting graphs, like finding an equation for points on a plane, as seen in this exercise. By mastering these concepts, students can apply exponential functions confidently in different situations.